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1 

2 

3 

4 

6 

6 

tllMlillMiMMW 


INVARIANTS  AND  EQUATIONS 


ASSOCIATED  WITH  THE 


General  Linear  Differential  Equation 


THESIS  PRESENTED  FOR  THE  DEGREE  OF  PH.  D. 


3 


■y 


r^; 


i*/^"^ 


GEORGE  F:  METZLER. 


JOHNS  HOPKINS  UNIVERSITV, 

BALTIMORE. 
I89I. 


s£m 


PRESS  or 

ISAAC    rRlBDBIIWAI.D   CO. 
■  AI.TI«0««. 


57^ 
T? 


Introduction. 

The  formation  of  functions,  associated  with  differential  equa- 
tions and  analogous  to  the  invariants  of  algebraic  quantics,  has 
occupied  the  attention  of  several  mathematicians  for  some  years, 
because  of  their  great  value  in  leading  to  practical  as  well  as 
theoretical  solutions  of  such  equations. 

Starting  with  the  work  of  M.  Laguerre  and  of  Professor 
Brioschi,  M.  Halphen,  in  two  important  memoirs,*  indicated  a 
method  for  the  formation  of  invariants,  but  involving  very  diffi- 
cult analysis.  He  derived  the  two  simplest  invariants  for  the 
cubic  and  quartic  and  such  derivatives  as  may  be  deduced  from 
them.     For  this  purpose  he,  by  means  of  the  transformation 

Y=ye  '•  ,  brings  the  equation  to  a  form  having  zero  for 
the  coefficient  of  the  second  term. 

Meauwhile  Mr.  Forsyth,  starting  with  the  letter  of  Professor 
Brioschi,  prepared  a  very  valuable  memoir,']'  in  which,  by  means 
of  the  following  transformations,  he  obtains  a  canonical  form  in 
which  the  coefficients  of  both  the  second  and  third  terms  vanish. 
This  may  be  stated  as  follows : 

When  the  linear  differential  equation 


(t)'-: 


^  P,  =  o 


*"  M^moire  snr  U  redaction  des  ^qaationrdiSerentielles  lin^aires  auz 
formes  integrables"  (M/moiru  du  Savanti  £tr»ngtrt.  Vol.  aS,  No.  i, 
301  pp.,  1880).  Also,  "  Sur  les  invarients  d«s  Equations  differentielles 
lintfaires  du  quatriime  ordre  "  [,Acta  Mcitk.,  Vol.  3,  1883,  pp.  3>5'-38o). 

f "  Invariants,  Covariants  and  Quotient  DerivatiTCS  associated  with 
Linear  Differential  Equations."— /VI*/m«/;Ii'm/  TrantaetiMu  •ftht  Royal 
Sttitty  of  Ltndtn,  Vol.  179  (1888),  A,  pp.  377-489. 


has  its  dependent  variable  y  transformed  to  «  by  the  equation 
V  =  «A  A  being  a  function  of  x  and  its  independent  variable 
changed  from  x  to  z,  where  z  and  A  are  determined  by 

dz 


3    —   *-»  —  > 


=  f-'. 


rfV 


+ 


rf;r 

/>  ^  =  o, 


dx*  ^  n+  I 
the  transformed  in  «  is  in  the  canonical  form 


(0 

(2) 


(/*» 


(t)e-£5.©c. 


d—*u 
dz'-*  ^ 


+    Qn  =  0, 


( — )  being  the  binomial  coefficient  ^pr^zTf] ' 

The  coefficients  P  and  Q  of  these  equations  are  ««  connecteji 
that  there  exist  «  -  2  algebraically  independent  functions  a  (^) 
of  the  coefficients  P  and  their  derivatives  which  are  such  that 
when  the  same  function  »,(*)  is  formed  of  the  coefficients  Q  and 
their  derivatives,  the  equation 

e,ix)  =  ij^y^aiz)  (3) 

is  identically  satisfied.    For  this  form  of  the  differential  equa- 


tion 


where 


r=<r— 8 


M^)=Q,  +  ^   ^£^  (-»)'«'. 


rf^g>-. 


<»r,»  — 


ff  —  I  U  -  2  1  2ff  -  .^Tlll 


Thus  e,{z)  is  independent  of  the  order  of  the  equat^n.  In 
this  z  is  completely  determined  by  equations  (0  and  (2).  But 
there  may  be  difficulties  in  the  way  of  solving  (2).  and  thus  it « 
desirable  to  form  the  invariants  for  the  uncanomcal  form  of  the 

***ForThis  purpose  Mr.  Forsyth  establishes  relations  between 
the  coefficients />  and  Q  for  tt»e  case  in  which  .  being  arbjt^ 
is  given  the  value  x  +  c;«.  where  .  is  so  small  tnat  the  square 


;  by  the  equation 
pendent  variable 
nined  by 

(2) 


.  +  Q,-o, 


I  are  so  connected 
int  functions  S  {x) 
lich  are  such  that, 
coefficients  Q  and 

(3) 
:  differential  equa- 

i!' 

the  equation.  In 
5(1)  and  (2).  But 
ig  (2),  and  thus  it  is 
nonical  form  of  the 

;s  relations  between 
har,  being  arbitrary, 
lall  that  the  square 


5 

and  higher  powers  may  be  neglected,  and  ii  is  an  arbitrary  non- 
constant  function  oix.    These  relations  are  expressed  thus : 


,  (nut  — 1  r  ,! 


{nis-o-i)  +  s  +  o-i}P>  J^_r+'i] 


(5) 


These  relations  are  fully  developed  in  Mr.  Forsyth's  memoir ; 
also  in  Dr.  Craig's  excellent  work*  they  will  be  found,  and  such 
a  general  treatment  of  the  whole  subject  of  differential  equations 
and  differential  quantics  as  makes  the  work  an  invaluable  help 
and  guide  to  any  student  of  the  subject. 

Then  we  derive 


d'Q. 
dsf 


dxr 


\i-{r  +  s-).!.'\-s.p/£^, 


m=i     imlr  —  m  +!!''■ 


tir-m  +  l// 


-  *=3i— 1  r         ,1 


(6) 


The  only  invariants  that  have  been  formed,  so  far  as  I  know, 
are  ^i ,  *« ,  *. ,  <'«  and  ^, ,  where  ^,  is  the  invariant  of  the  rth  order 
of  an  equation  of  order  n. 

In  Section  I  of  this  thesis  the  general  invariant  0,  is  con- 
sidered, and  it  is  there  shown  that  in  the  non-linear  part  every 
term  is  of  the  form  ABC.  Where  ^  is  a  number,  ^  is  a 
function  of  P,  and  its  derivatives,  and  C  ?>  L\n  invariant  or  the 
derivative  of  an  invariant  with  suffix  diffennf  from  s  by  an  even 
number.    When  s  is  even  C  may  be  a  number. 

Section  II  deals  with  the  coefficients  of  (f„  giving  some 

*  Treatise  on  Linear  Differential  Equations.  By  Thomas  Craig,  Ph.D. 
Vol.  I. 


)i^4 


general  expreMtons  by  which  they  may  be  calculated  for  any 

value  of  i.  . .         J  • 

Section  III  treats  of  associate  variables  and  associate  equa- 
tions, showing  which  are  identical  and  which  may  not  be. 

Dr  Craig  having  discovered  that  the  condition  for  the  self- 
adiointness  of  the  sextic  and  octic  was  that  their  invariants  with 
odd  suffix  all  vanish,  suggested  to  me  the  general  theorem 
announced  in  his  treatise,  pp.  293-295.  The  proof  given  at  that 
time  only  applied  to  equations  in  Mr.  Forsyth's  canonical  form. 
By  aid  of  what  is  established  in  Section  I,  it  is  shown  to  apply 
also  to  equations  in  any  form.  ...  ,    , 

A  fuller  history  of  the  subject  will  be  found  in  the  works  to 
which  reference  has  been  made.  ^    ^    .     .      .      u  j 

This  paper  was  not  only  suggested  by  Dr.  Craig,  but  has  had 
his  valuable  criticism. 


Iculated  for  any 

i  associate  equa- 
nay  not  be. 
tion  for  the  self- 
ir  invariants  with 
general  theorem 
roof  given  at  that 
s  canonical  form. 
;  shown  to  apply 

d  in  the  works  to 

>aig,but  hashad 


-MMM 


MM- 


" 


Section  I. 

The  Form  of  the  General  Linear  Prime  Invariant  t*,. 

Since  (i,  has  only  a  linear  part  when  Z',  vanishes,  its  form  must 
be  as  follows ; 

[AP.  4-  BF..,  +  (:/>:'-,+  .  .  .  +  fWi-"  1 

+  [/»,  {a.«._,        +  «.^.- .  +  «i^'-.  +  •  •  •  +  «.-.A-*'  I  ] 

+  in  \  *.«.-»  +  *4K  -.  +  ...  +  ^  -.''i'""  { ] 

+  [p';s  ^A-«  +<:.<*:-.+...  +f.-./'i—'n 

+  [P\)  rfA-4  +</.»:-.+... +''.-,/'i'-/'(] 

+  \.K\ \ 

+  [etc J 

+  [etc ] 

etc.,  etc 

In  this  (r)  is  the  differential  index,  so  that 

The  sum  of  the  suffixes  and  differential  indices,  it  will  be 
noticed,  equals  s  for  every  term ;  that  is,  B,  possesses  a  kind  of 
homogeneity.*  s  is  called  the  index  or  dimension  number  of 
0,;  the  dimension  number  of /'J"'<T-«  being 

Denoting  the  terms  within  the  square  parenthesis  by  L,  «,  ^,  r* 
i,  etc.,  then  »,  =  Z,  +  «  +  ^  +  r  +  «  +  • .  • 

The  notation  used  here  will  be  nearly  that  used  by  Mr. 
Forsyth,  but  to  simplify  the  work  the  m's  and  their  derivatives 
arising  from  b=ix  +  sii  will  be  dropped,  that  is,  they  will  be 

«  Philoaophical  Transactions,  Vol.  179  (1888)  A,  pp.  391-92. 


6s 


KM 


mmmm 


8 


treated  as  unity,  when  the  result  will  not  be  changed  by  doing 


so.     Also  Z^"'  =  -J  I  will  be  considered  =  with  P, 


doP 
The  general  form  of  the  terms  in  Z.  is 

I  shall  now  show  that  when  *  is  odd  each  of  the  numerical 
coefficients  a,,b,,c,,dn,  etc.,  of  the  non-linear  part  of  «,  equals 
zero. 

From  page  4  of  the  introduction  we  have 

identically  satisfied.  If  in  the  right  member  of  this  identity  the 
Q'i  and  their  derivatives  are  replaced  by  their  values  in  terms 
of  the  /"s  and  their  derivatives,  as  expressed  by  formulae  (5) 
and  (6)  (page  5,  introduction),  then  the  terms  of  dimension  '  s ' 
in  each  member  cancel,  those  of  dimension  '  j  —  i '  furnish  the 
numerical  coefficients  in  the  linear  part  L,  and  there  remain 
terms  of  dimension  equal  to  and  less  than  s  —  2  with  which  we 
may  determine  the  coefficients  of  the  non-linear  part. 

Remembering  the  convention  Pl"^  =.  /*,,  formulae  (5)  and 
(6)  are  included  in 


^•  =  /'i'>ii-(r-Hj)./i'| 


d^ 


r  =  o,  I,  2,  3  . . .  i 


(7) 


Also,  difTerentiating  the  invariants,  we  find 


anged  by  doing 

i,2....f— 2.  (a) 

>f  the  numerical 
part  of  ff,  equals 

this  identity  the 
r  values  in  terms 
by  formulae  (5) 
of  dimension  '  s ' 
—  1 '  furnish  the 
nd  there  remain 
2  with  which  we 
T  part, 
jrmulae  (5)  and 


+  1) 

r-m  +  l) 

1) 

•  +  »)( 

r) 

t,  3- 

.  S 

■    (7) 


i 


9 

Then  i^.  =  /».  (I  -  2t//)  -  ?  +  '  «//", 

4 

I?,  =  /»,(!-  6t/i')  -  i5e//'/»  -  4  («  +  9)  ^m'"/** 

-  5  («  +  4)'/^"'/'.  -  *(3«  +  7)  '/^'•"/'.  -  "  -;^  "  5*/^"*. 

»4 

'g  =  />;  (1  _  3.//)  _  2.m"/>  -  "-±i  c/.'^ 

-^^  =  />i"  (I  -  6*//)  -  gei/'P:  -  ^^ti."'P, 

- « ^  )6jr«"iP.  +  (« +  5)  m"'^.{  -  a  (« +  0  *M^ 

etc. ...     etc. 

From  these  follow 

iK  =  /»; )  I  -  2r./i'}  -  r/>;-'  "J  ^  .m'«. 


I 


} 


lO 


If  /^,"«*i*_-«"  be  a  term  in  S.,  then  will  the  term  -^       ^^I, 
be  multiplied  by  (/)'  or  (i  —  siit),  and 

=  (I  +  «y)  I  P',"  (I  -  seja')  -  9«m"^.' 

«=«-•  r        X  —  5! 

|(.-x)(.-4-»») +»f  i^i-'.M'"-*— *]• 

In  this  equation  the  terms  oif  dimension  '  s '  cancel  and  —  e 
is  a  factor  of  the  remaining  terms,  so  that  when  every  term  in 
S,  is  treated  in  this  way,  all  terms  of  dimension  '  s '  cancel  each 
other  and  the  remainder  is  divisible  by  —  t.  Denoting  by  HL 
the  remainder  of  the  linear  part  A  by  H,  the  remainder  of  the 

(ti  \  **  • 

— j  the  binomial  coefficient  jp(:^zry\' 

also  omitting  the  /x's  and  dividing  by  —  e,  we  get 

jiL^Al"^  />._.+  '-^.in+  I  +  2is-  2)/'._,  +  etc.1 

L     "*  2.3.  -J 

+  Bis- I  P.-1  +  .'•}+'•' 


...^'r.l^'irt.j-'oTfaifl; 


II 


1?        rf«— 


;»  L»*'«— 4"""*' 


e/t" 


■^ 


X  —  4  —  »» ! 


J '  cancel  and  —  e 

den  every  term  in 

)n  '  * '  cancel  each 

Denoting  by  RL 

i  remainder  of  the 

n\ 
efficient  yrf^ZTrl' 

iget 

f-  2)/»_, +  etc.J 

•  "J  T    •  •  • 

,.    x=l,2,3...* 


+  C [{s-2)P,-^  +  ...  +  Y7^{  2 (j  -  2)  +  ij  />;_, 

+;i:{r^)"-^'S-(s)}(^-'' >" 

for  the  first  three  terras  of  B,.    Replacing  A,  B,  C,  etc.,  by  their 

s      s.s —  I  .s—  2 


r—i 


(lO) 


) 


values  ((a)  p.  8)  I,  -  -|^ ,  ^"^   •/_  ^   ,  etc.,  the  (r  +  i)9t 

term  gives 

,_    y      s\s—2\2s—r—2\ 

^      ^'  2. r!  J— r!  J— r — i !  2^—3! 

f  (s  —  r\  n  +  I  X  —  r~  I 

\v-r)      2      X  —  r+i 

x  =  r  +  i,r  +  2, 

as  a  remainder.  By  giving  r  all  values  o,  i,  2,  3, ...  (10)  ex- 
presses the  whole  of  HL. 

*«=  P,  [J  (*-  3)  «A-.  +  «*)*-4.<'.-«  +  (2*-7)<^.-*l  +  etc.] 

r  =  X  —  2  J  , 

with  similar  expressions  for  the  other  parts,  Urt  R^,  etc.  Suppose 
that  the  coefficient  of  /»<!»,  in  RL  is  A^(n  +  i)  +  A  +  C 
Then  the  general  forms 

(->'T  iimi'^XB^)  ^^  <" ^-'"'^ 

r  =  »,  w  +  I, ...  X  —  I, 


Siissii; 


12 


(-.)' H>-)(^Fi-^-)(r^-.>"-'-'"' 


r.=  w, »  +  I,  •  .  .  X  —  I. 


and 


(-.)-(t)&E1)(^^5^)[(t)(«-')-(.^t)] 

will,  when  expanded,  give  A^,  B^  and  C  respectively.  In  these 
i^l.-AZL^  is  the  reciprocal  of  f^^^).  Thus  A  is  found 
to  be 


^.=(-.)-(-:-)(SXii^)r»[ 


L  fz^— »>— 


2...  2f — X  — 2 


X  — W+I 


X— t;— I 


X  —  V\ 


S  —  V  —  I 


+  25-z>-4---2Y'~''='-z^-3 


j  —  t; —  I  .  s  —  V—  2 


X —V—l\  '         ^  1.2 

2S  —  X  .2S  —  X  —  I  .  .  .  2S  —  X  —  2 


(-1) 


X  S-V-l...S-x+_2   ^   ^^^ 
^  X  —  V  —  2! 


•] 


\  (S  —  X  +  V  —  I  ...  J  —  I 

j  -  ^  [ 's-v+i\ 


S  —  V  —  1  .  .  .  S  —  x\ 
X  —  v\ 

S  —  V  —  I   .  ,  .  S  —  X 


X  —  V  —  I 


Use  the  upper  or  lower  signs  according  as  x  —  » is  odd  dr 
even.    To  obtain  this  result  expand 

«* (I  —  ;r)— — '  =  jc'-Cs  —  v-  i)x' 


+ a «*-... -(-I)        i,_t,_2l^ 


-} 


(a) 


mmmmmmmm 


^t  —  »l     I 
+  1,  ...  X  —  I, 

lively.    In  these 
Thus  A  is  found 


<S—X—2 

n 

V  —  I 


X  —  V—l 


V—  I  .  S  —  V—  2 
~       I  .  2 

—  X  —  2 


S—*  \-V  .  .  .  ^— I 
X  —  v\ 

.  .  .  S  —  I 


-i! 


13  X  —  3;  is  odd  dr 


•v—a\' 


,..}, 


w-ii^_.  ?-(») 


'SiB^irWriini*  wnnw  wiwiMni 


13 


and 


.j_,v-.-.)  ==  ^  +  (25  _  X  _  2)  -i 


2f 


X  —  V  +   112S  —  X  —  2\ 

Differentiating  the  last  equation, 


—  2X~ 


•(I   -AT)-"-'-" 

_  (2*  -  X  -  2)  X-*  (I  -  *)-<»'-'-' 


_  —  I   _  25— X  — 2 

+  .  .  .  x—V—l 
+   ... 


+  0  + 
2S 


2S 


3 !  2  J  —  X  —  3 ! 

V —  2  ! 


—  t/+i !  25— X— 3! 


(b) 


The  coefficient  of  Jf""*  in  the  product  of  the  right  members 
of  (a)  and  (b)  is  the  series  of  terms  in  square  parenthesis  in  the 
expression  of  Ai  above,  and  the  coefficient  of  x"—  in  the  pro- 
duct of  the  left  members  is  the  quantity  within  square  paren- 
thesis in  the  final  value  given  for  A^. 

Bx  is  found  by  putting  (i  -  x)— — *  and  (i  -  xy—'-*  equal 
to  their  expansions  and  taking  the  coefficients  of  *"-•+*  from 
the  product  of  the  left  members  and  also  from  the  product  of 
the  right  members.    Then 

^^~^~^^''\^)\v^)\2S-i)  ~2V    [_~~         X-V+  l~~ 

If  in  these  expressions  fory4i,  Bi  and  G,  »  is  made  equal  to 
zero,  then  for  all  odd  values  of  x 

Ar  =  o  =  Bx  +  Q,  (II) 

while  for  even  values  of  X 

."^a^^/Ut^/  I  („) 

j^______^  („  +  I)  + ___       J  j 

For  v=  I 


A,  («  +  i)  +  ^,  +  Q 


H 
and  X  increased  by  unity,  ^i  («  + 1)  +  ^i  +  Q  becomes  the 

same  as  in  (12)  multiplied  by ^  .    Then  in  HL,  if  JFbe 

the  coefficient  of  Z'.  _ ,  when  x  is  even, 


-  W. 


S  —  X 


is  the  coefficient  of  ^_«_i. 


(13) 


W hen  v  =  x  —  2,\etAi(n+i)  +  Bi+Cihe denoted  by  a„ . 
The  following  are  the  values  of  -<4, ,  Bi  and  Ci  when  » = x  —  2  J 


^'-^-"'^   [^)\x-i]{2S-3)  J-xfl.J-x.4 


2J— X— I  .  2f— X— 2  .  2J— 2X  +  3.  x—i 
2.6  ~ 


'■.  =  (-)-'(TX£fXiI^,) 


2S—X—2.  y— 2«  — 2.x— I 
2.6  ~ 


Now,  when  the  whole  remainder  is  considered,  the  coefficient 
of  each  of  the  (/'i*2j^)'s  must  be  zero.  Let  us  now  consider 
those  terms  of  dimension  x  —  2.    They  will  be  foiind  only  in 

^     If 

XL  and  /?a.    The  coefficient  of /*._,  is  —  oj,  +  —J—  «i.  This 
equals  zero,  and  when  v=o and  x  =  2 

«■■  =  i  (l)('-^)(sii)««  -  3X»  +  ■)  +  ••  -  5-  +  61. 
therefore 

The  coefficient  of /^_,  is,  by  (13). 


— V —  a,  +  — ^—  tf» 


*—  2 


2 
a,  =  o. 


o,,; 


«+  1 


o*; 


dien 
The  coefficient  of  P'.Lt  is 

6     *  ^    6     'V  2  y  4 .  2*  -  3 


i;,.ijpi^iiiSi4..^''!:Wi'./iM^'.^^.  .•'■  ■ 


WTBilMliiiii  ni'\t^'«  ttiMOBMimtamfi^aM.'m 


Q  becomes  the 
iiin/?Aif  IVbt 

-1-  (13) 

e  denoted  by  a^. 
,  wheni'=*  — 2: 

.  2S—*—l.2S~x 


2S—2*  +  $.x—l 
—  2«— 2.x— I 

.6  ""• 

'ed,  the  coefficient 
i  us  now  consider 

be  foiind  only  in 

.  n  +  i 


Ot.  This 


+  J»-5*  +  6}. 


+  (f-2).(i-3)}. 


«  +  I 


a»', 


-3 


—  flw 


J 


15 

Substituting  for  Ot  and  «,« their  values, 

«•  =  J^  (i)(-l  ')  ^  (-'"^  .X..-5)+.-4..-5}. 

Calling  the  three  terms  whose  sum  gave  the  coefficient  of  P','_^ 
i,  II,  K,  then  the  coefficient  of /*i?.,  is 

"^  a,  +  -'^  -I  +  -f~4ri-^  "  -  «»  =  "»  +  ^1  +  /^.  +  ''i. 

say.    The  last  three  terms  reduce  to  zero ;  therefore 

a,  =  0. 
The  coefficient  of /*i*i,  =  »,  +  ^»  +  /^i  +  «i.,  say 

s  —  ^.s  —  6 


4 .  2J  —  9 


.«!—   «1.  =  0. 


Reducing  this, 
*•  =  ='^i^  \i-6!j-6!2J-3!3!2 


—  6    /       *!.r— 2!  2f  —  12!       \,   ,     ,     X,  , 


+  s  —  6.s  —  'j\. 


Similarly  th  may  be  shown  equal  to  zero  and 

—  6  j!*  — 2!2*— 16!         ,   ,     ,     ..  -. 

«•  =  ^+7  2.3i^-8i.-8!2.-3l  ^^^"  -^  ^^^"'  -  9^ 

+  5  —  8.  J  —  9}. 

Had  the  terms  in  the  coefficient  of  PSii  been  denoted  by 

*4-^  Ot,  a,,  A,,  Ai,  and  Ox,,  then  those  giving  a,  would  be 
o 

6         •         2.2J— 15      *         4.2J— 13     •         *    


6 .  2*—  1 1 


It  thus  appears  that  A.,  ju,,  ct  have  a  relation  between  them 
similar  to  >l„  /nt,  r*  and  K%ih,«rx,  etc.,^and  if  we  follow  the  same 
law  the  coefficient  of  i^ri  becomes 


+    1  ,/  Ml  +  lf  *U— 2I2J— X  — 2/— 2! 

2J— 2x.2f— 2X— 2.i9fl,      , 


i6 
where  ^  =  2*  —  4/  —  i  and 

0,  =t  {/>(«  +   0(2*  -2p-l)  +  is-  2P){S  -2P-  l)K 

2/  =  2,  4,  6  ...  X  —  I  or  X. 
Also  a,  would  equal  zero  when  x  is  odd,  and  when  *  is  even 

*"      «+  1  2.3l5-x!5-x!2j--3!  V2  '^ 

To  prove  that  this  law  holds,  consider  the  series 

/  =  (-  I)"  [(«  +  I)  4T7-7rr:rx -TT 


•  —  2 !  2*  —  X ! 


!  X  —  2 !  2J  —  3 1 


J !  J  _^  zr—  X  —  1 !  (25  —  2«)(  J  —  2x  +  3) 
^rr—x !  5  —  x—  i!x  —  2!2J—  3! 

^ !  ^  —  2 !  2.?  -  X  —  2 !  (3f  -  2x  -  2)(2.y  -  2x)(s  -  X  +  i) 

4!^  —  x!5  —  x+  l!x  —  2!25— 3! 


5  !  J  —  2  !  2J  —  X  —  2^  —  2  ! 


2S  —  2X 


4!x  —  2^!*  —  x!j  —  x  +  i!2*~3! 

X  2J  —  2X  +  2  .  2J  —  4^  —  I  .  <?/, 

2/  =  2,  4,  6  ...  X  —  1  or  X. 

The  first  three  terms  are  what  A .  ^1  and  Q  become  when  v  is 
made  equal  to  x  —  2.    As  the  series  is  to  be  shown  to  be  equal 

-  .  S\S—2\2S  —  2X  —  2 1 

to  zero,  the  common  factor  (—1)"  a\s  —  x\s  —  x  +  it  2s  —  ^i 
may  be  omitted.    Then 

-;?• .  2J  —  X  —  /^  —  I  .  .  .  2S  —  2X  —  I 

77^^+ 2! 


/2^  — x-jg-\  _  25  - 
\x-g+2J 


2S-.-jr    2S-.-^-l  ^  (^  +  2)  =  ^  (^).         (X4) 


x—g+  2.x— g  +  1 

The  series  to  be  considered  now  becomes 


m 


■MMi 


mmmmtm 


% 


dwhen  *  is  even 

){2S  -x-l) 

ieries 

xj^ 

2 !  2  J  —  3 ! 
0 


2X){S  ■ 

—  X 

+  1) 

-3! 

, 

S  —  2X 

.25  — 

4/> 

—  I. 

ep 

'i  become  when  v  is 
shown  to  be  equal 
2 1  2J  —  2«  —  2 1 

r  — X  +  I !  2J  —  3 ! 


.  . .  2f  —  2X  —  I 

.2;ir(^).  say. 
i)  =  x(jr)'     (14) 


17 

/  (4)  .  25  —  X  .  2J  —  *  —  I  .  2J  —  X  —  2  .  2J  —  X  —  3  .  («  +  l) 

—  ;f  (4)  .  25  —  X  —  1 .  2J  —  X  —  2  .  25  —  X  —  3  .  25  —  2x .  5  —  2x  +  3 
+  ;f  (4)  .  25  —  X  —  2  .  25  —  X  —  3 .  25  —  2x  .  5  —  X  +  1 .  35  —  2x  —  2 

—  ;|r(4).25—  2X.25— 2X  +JI.25— 5(n(25— 3)  +  5*  — 35  +  3) 

—  ;f  (6)  .  25—  2X  .  25  —  2X  +  2 .  25  —  9  (2« .  (25  —  5)  +  5*  -  55  +  lo) 
— /(8).25— 2X.25— 2X  +  2  .  25  — 13  (3«.  (25— 7)  +  5*— 75+21) 


— /(2^+2).25  — 2X.25— 2x  +2.  25  — 4/— l{/»  (25  —  2/— l) 

+  5»  -  (2/ +  1)5 +/(2^  +  I)}, 

2/  =  X  —  I      or     X. 

Consider  the  coefficient  of  n, 

;^  (4)  [25  —  X .  25  —  X  —  I  .  25  —  X  —  2 .  25  —  X  —  3 

—  25  —  2x  .  25  —  2X  +  2  .  25  —  5  .  25  —  3] 
= ;?« [8^  —  4*  (2«  .+  3)  +  *  (*  +  11)]  X  —  2  .  X  —  3 

=:x  — 2.x  — 3./(4)  Ji.say, 

=  25  -  X  ^  4 .  25  —  X  -  5  Jvir  (6)  by  (14). 
Take  from  this 

;f  (6)  .  25  —  2X  .  25  —  2x  +  2  .  25  —  9 .  25  —  5  .  2i 

and  the  second  remainder  is 
;f(6)x-4.x-5.[i25»-65(2x  +  5)  +  x(x  +  29)] 

=  X  —  4. X  —  5  .;f  (6)  4„  say. 
This  equab 

25  —  X  —  6 .  25  —  X  —  7 .  ;if  (8)  J,  by  ( 14). 
Take  from  this  the  next  term  of  the  series, 

;f  (8) .  25  —  2X  .  25  —  2X  +  2  .  25  —  13 .  25  —  7  .  3 ; 

the  remainder  is  " 

x-6.x-7;f(8;[l65»-85(2x.+  7)  +  x(x  +  55)] 

=  25  —  X  —  8.  25  —  X  —  9 ./ (10)  J,,  say. 

Supposing  this  law  to  hold  for  all  differences  till  the  (m—  i)th, 
it  can  be  shown  to  hold  for  the  mth.    The  (m  —  i)th  is 


i8 

X  +  2  -  2W.X  +  1  -  2«.;^(2«)[4W5*-  2»m(2x  +  3«  -  l) 

+  X  (x  +  4W'  +  aw  -  I)]  =  z(2«  +  2)  ^—1 
2j  —  X  —  2»» .  2J  —  X  —  am  —  I. 

Taking  from  this 
there  remains 

-  2  (W  +  l)(2x  +  2W  +  I)  +  x(x  +  4«'  +  6»»  +  I)] 

=  «  -  2m .  X  -  2WI  -  lAf  (2»»  +  a)  J, ; 
that  is,  the  mth  difference  is  the  same  function  of  m  as  the 

/^-,_  i)th  is  of »»  —  I.  ..     ,     ,    ,^    _ 

When  2W  =  2/  =  X  -  I  or  X  the  subtrahend  is  the  last  term 

of  the  series  and  the  difference  vanishes.  ,  Thus  we  see  the 

coefficient  ofn  in  the  series  vanishes.  .»/•«;. 

The  algebraic  sum  of  the  first  four  terms  mdependent  ot  n  is 

X- 2.x-3;?(4)[2^--**(2x  +  8)  +  icw(x  +  1) 

then  by  (14)  it  equals 

Jj2«  —  X  —  4.2J  —  X  —  5.  ;ifC6). 

Taking  from  this 

;^(6)  2J  -  ax  .  2J  -  2x  +  2  .  2*  —  9  •  *•  -  5*  +  1° 

there  remains 

„.6)x-4.x-s[a**-(«  +  «)^  +  (i4«+25)*-«(«  +  29)] 

If  the  (m  -  i)th  difference  be 
x-2».  +  2.x-aw  +  i./(a«.)[ai*-(ax  +  4«)^+{2«(2«+i) 
+  2(m  -  i)(2i«  +  i)}*-x(x  +  4»»»' -  2*«-  ^>J' 
which  we  will  denote  by  <^  («  -  1) ;  then  the  »ith  is 

<&(«- i)-z(2'«+2)[2J-2x.25- ax  +  a.  a*- 4»»- I 

=  y  (aw  +  2).x  -  2m. X  -  2m  -  i[2«»-  i*(ax  +  4(»»  +0 
4-{3x(2m+3)  +  2»»(2'"+3)}*-«(«+4»»*+6»»+0J 


^i^,i»ig9E.. 


J  (2x  +  2m—i) 

(2IW  +  2)  -^.-i 
I. 

--1.2S  —  /^m  —  t. 


+  4»»'  +  6»t  +  i)] 

ction  of  m  as  the 

nd  is  the  last  term 
,  Thus  we  see  the 

ndependent  of  n  is 

t  +  i) 

2.x  —  3Xt^i'^^y> 

(6). 

t»  -  54  +  10 

»-25)i-x(*  +  29)] 

|»l)j»+{2x(2lf»+l) 

+  ^m*  —  2m-  i)]. 
tie  mth  is 
+  a .  2*  —  4***  ~"  * 

l)j  +  «»(2»»+l)}] 

a*  (2x  +  4  (»»  +  i) 
«+4w*+6»i+i)] 


»9 

This  vanishes  when  2m  =  x  or  x  —  i,  and  also  completes  the 
series. 

Thus  the  whole  series  has  been  shown  to  vanish  whatever  be 
the  value  of  x.  (15) 

Assuming  that  <!«  =  o  when  x  is  odd,  and 

—  6  *!j— 2!2J  — 2x!  f    X    /       ,       x/  \ 

=  W+i2.3\s-.\s-.\2S-3\    |y("+0(2^-x-1) 

+  (f  -  x)(f  -  X  -  I)  } 

for  all  even  values  of  x  less  than  221/  +  i>  then  it  may  be  shown 
to  be  true  when  x  =  2w  +  i  and  2W  +  2.  The  coefficient  of 
/'i-tolii  in  HL  is  a,„+ti  and  if  Mi  represent  the  value  of «« 
when  X  is  even,  and  N^  represent  the  expression 


(-1)^ 


SIS  —  2\2S  —  T 


2\t\s  —  TlS  —  T  —  l\2S—  ^l' 

i.  e.  the  coefficient  of  PlV.r  in  L,  then  the  whole  coefficient  of 
Pi*:^ii'U  is 

^-  a^+,  +  «,.„+,  -  iM\Nf-V  +  MINT--,*  +  MvNtLV 

+  . . .  +  i«/;jNri_..]  ^^ 

Now  <h.*m+\  is  the  sum  of  the  first  three  terms  of  F,  and  the 
following  terms  are  those  of  Talso;  for  taking  any  one  of  them, 
as 

it  becomes,  when  written  in  full, 

6     «+i        s\s  —  2\2s  —  4^ 
«  +  I      ( 


2.3u-2^u-2^!2.:::i!  ^*(«+')(«-^^-») 

-'+(s  —  2g)(s  — 22— i)] 

S  —  2gl  S  —  22  —  2  \2S  —  2W—28  —  Z 

2\2W  —  22  +  l\  S  —  2W  —  l\  S  —  2W  —  2\  2S  —  ^  —  ^\ 

S\  S  —  2\  2S  —  2W  —  22  —  x\  .  0, 

=s  — , , i 3 5 f  2S  —  AW 

\\  2W  —  22  -^  l\  S  .  2W  —  l!*— 2ZC!2f— 3!  ^ 

X  2J  —  4W  +  2  .  2J  —  4«  —  1, 


d 


[ill  ■  1 

II  ! 


PI 


i  ,;i 


90 
which  coincides  with  the  last  terms  of  1'  when  «  =  aw  +  i  and 
m:=/>.    Thus  the  coefficient  of  P!^i.t  i  consists  of  ^^-g—  «..  + 1 
plus  a  series  of  terms  which  vanish  by  (15) ;  then 

The  coefficient  of  /*!!!'», -1  is 


(16) 


+  . . .  +  ^;.^.'-i-]  =  o- 

r  gives  all  the  terms  in  this  expression  when  x  =  2a/  +  2, 
excepting  the  first  or  "^  a,.+..  But  the  last  term  M\J^]-,. 
is  the  second  last  in  F  when  x  =  2a;  +  2,  2/  =  2, 4  •  •  •  aw  +  2. 

Taking  T  from  the  above  coefficient,  -g-  Of+i 5-  ^•''+« 

is  the  coefficient,  since  r  =  o  always.    And  as  this  must  vanish. 

Thus  (16)  shows  that  if  for  any  odd  value  of  «  and  all  lower 
odd  values  a.  =  o,  then  «.+.  =  o.  and  (17)  shows  that  if  for 
any  even  value  and  all  lower  even  values  a,=  M'n,  then 

««  +  !  =    J^'n  +  t' 

On  pages  14  and  15  it  is  shown  that  «.  =  o  for  x  =  3. 5. 7  and 
a  ^;  for  X  =  2,  4, 6.  8.  Therefore  it  follows  that  (16)  and 
(17)  are  true  for  all  values  of  o'. 

It  foUows.  then,  that  in  9,  the  row  of  terms  designated  a,  of 
which  Pt  is  a  factor,  contains  no  invariant  or  derivative  of  the 

This  is  also  the  case  for  the  terms  entering  in  the  row  desig- 
nated /9  and  of  which  Pi  is  a  factor,  for  the  term  /\^.-4  is  found 
only  in  Ra  and  Hfi.    Its  coefficient  is 

2*4  +  (2J  — 7)<»«; 

then 

A  _       g^-7  - 


ai 


X  =  2W  +  1  and 

tS  OI  — z —  <»iif  +  l 


tien 


(16) 


hen  x  =  2w  +  2, 
=  2, 4  ...  aw  +  2- 

}  this  must  vanish, 
C17) 
[>f  X  and  all  lower 
I  shows  that  if  for 
=  J/;,  then 

>  for  X  =  3, 5. 7  and 
3WS  that  (16)  and 

ns  designated  a,  of 
:  derivative  of  the 

(18) 

J  in  the  row  desijj- 
irm  /».^.-«  is  found 


Any  term  as  /'■^i-';",  *  being  odd,  could  appear  only  in  i?a 
and  i?/9,  and  as  it  does  not  appear  in  Ha  it  cannot  in  H^, 
The  coefficient  of  Pt^ir^i;"  is 


or 


a*„  +  («  -  i)(aj  -  a«  -  3)  a»,  =  o.  (19) 

The  terms  of  dimension  s—  i  and  of  form  Pt'^Z,  can  appear 
only  in  7P/9  and  Hy,  and  when  *  is  odd  no  such  term  appears 
in  /?/9 ;  therefore  it  does  not  enter  into  Xr. 

When  X  is  even,  the  coefficient  of  /'i^i-M*'  is 

»'-M(i^)<'-">-(s^)}'"=°' 

or 

5^,.  +  (2K  -  3)(5  -  «  —  2)  *„  =  o.  (ao) 

In  this  way  it  is'  easy  to  see,  by  taking  one  row  after  another, 
that  the  non-linear  part  ofB,  contains  no  term  having  0i,'lK  as  a 
factor  when  x  is  odd.  (21) 

From  this  it  follows  that  if  all  the  invariants  of  a  diflferential 
equation  with  even  suffix  vanish,  the  linear  part  of  each  vanishes. 
The  same  is  true  for  those  with  odd  suffix.  (22) 


I 


mmmmmm 


Section  II. 
Thb  Coefficients  of  B,. 
tf,  has,  as  we  have  seen,  a  linear  part  expressed  by 


rmt  — 


"f     NlP^r, 


Then  follow  a  series  of  terms 
expressed  generally  by 


(24) 


6       —  [Sr]  j|j-alaj-4«i  ) 

{x(«+iXaJ-2«-0  +  ('-2x)(*-a*-i)}»i!li;") 

'nil  meaning  the  greatest  integer  in  '-^ .    Then  follow 

F>,  {dA-i  +  *.*"-•  +  *•**-•    +  -"^ 
+  Pli  {cA-4  +  (**>"-*  +  c^sY-,    +  . . .} 

+  /n"  {efi>.-,  +  *.«i'^.  +  «..*r-»  +  •  •  •} 
+ ••  V 

These  are  expressed  generally  by 


Hy.e^fi."-*'  +  /'i'-'"        2:"  "  q^B^tu'-*^ 


=  a,  4, 6  . .  •  etc. 


riBiMli 


mmmmmm 


i9MMrffNNHMi«HNHM||iH 


ised  by 


-3 

..  +  ...} 


P':ir.    (as) 


(24) 


Then  follow 


+  ...} 
+  ...} 
+  .  ..} 


-•] 


=  2,  4,  6  . .  •  etc. 


23 

If  any  two  coniccutive  rows  be  considered,  for  which  (w  =  /*), 
the  remaii'.der  arising  from  them  will  contain  a  term  ^ 

found  nowhere  else,  because  all  rows  preceding  these  have  /T' 
as  a  factor  where  v  <  n,  and  rows  following  them  have  a  re- 
mainder in  which  the  index  of  *,_„  cannot  be  as  great  as 
(ax  —  At  —  3).    This  remainder  is 

r*~'i 

+  ^/>jM-.)  + . . .  +1  "r   «„<><•==•'-" 

.«..["-r%-r{(^>-">-(;i^)}«-- 

r=  2*  —  fi—  2 
+  . . .  +  terms  of  lower  dimension  1       £  ^   ^w^i-wc"    " 

.^.^.{-r^/:r{(^)(.-.).(^)}«r... 

r  =  ax  —  M  —  3    • 

Equating  the  coeiBScient  of  the  term  Pir^eirLu"'*^  to  zero  we 
obtain 


1 


. 


•vr""-"."**™ 


24 

In  this  X  is  any  number  and  fi  any  of  the  values  of  v,  so  that 
the  coefficients  g^  of  any  row  may  be  expressed  in  terms  of 
those  of  the  preceding  row,  viz.  ««. 

(25)  when  simplified  gives 


/»+  I 


(4  +  f^)9 


_         (2X  —  M  —  2)(2S  —2x  —  ti—  3)  ■■ 


Making  m  =  o,  i,  2,  3  •  •  •  this  gives 

4. 1  .*,.  =  -  (2X  -  2)(25  -  2x  -  3)a„ 

5 .  2  .  f«  =  -  (2X  -  3)(2J  -  2*  -  4)  *« 

6.  3.  if„  =  —  (2x  -  4X25  -  2x  -  5)f„ 

•       •• »..••• 

■ 

(m+i)(4  +  /')?.«  =- (2«-/*-2X2*-2«-/*-3)«««- 

Equating  the  product  of  the  right  members  to  the  product  of 
the  left  gives 

ft+l!M  +  4l        (      ilu^i         2X-2J25-2X-3!  ^^g^ 

?«• J\  '^        ■'         2x— /I— 3  !  2*— 2x— /i— 4  1 

The  ^'s  being  coefficients  in  the  row  multiplied  by  PiT+J^  it 
is  seen  that  the  coefficient  of  any  term  of  the  form  Pi«'»i*ii*-" 
may  be  expressed  in  terms  of  the  a'a.  Writing  this  coefficient, 
for  brevity,  W^"*-" ,  then 


2x  —  2 !  2f  —  2X  — 3!fiJ  — 2!2J  — 4x' 


"{<() 


d\d  +  ^\2x—9—2\2S—2x  —  S—i\2S-3\s—2x\S—2xl2 

6 


'  •  (27) 


There  still  remain  terms  of  the  form 

Here  a,  6,  c,  d,  etc.,  are  indices  expressing  powers  of  the 
factors  to  which  they  are  attached.  (a*^^«Oir'  » the  coeffi- 
cient of  the  term  having  such  indices,  powers  and  suffix  s  —  2x. 


values  of  v,  so  that 
ressed  in  terms  of 


2x  —  /*  —  3) 


-  3) ««« 


«.« 


J2J— 2x— /t— 3)«»ic. 

rs  to  the  product  of 


""3'  ^.a>..  (26) 

tiplied  by  Pr+"  it 
e  form  Pi«'»i*^*-" 
ting  this  coefficient, 


IM. 


IxlS — 2x!  2 

6 


'  •  (27) 


«+  I  . 


')iJ'«i*u. 


sing  powers  of  the 

■•i*e')ir'  is  the  coeffi- 
•rs  and  suffix  s  —  2x. 


as 

Throughout  the  whole  invariant  the  order  of  the  factors  will  be 

taken  so  that  _    _    _    _ 

«<)?<>'<*<«.  etc.  (28) 

2x  =  »»  +  a(a  +  2)  +  *(/S  +  2)  +  tf(r+2)  1       (^q) 

+  rf(a+2)+<f(«  +  2)+  ...  J  ■    ^"^ 

The  numerical  value  of  (a'/S»r°^cOir»  is  found  by  equating 
the  coefficient  of />i"'"-PiP''/'r"/*i*''/'i"*"*^i^«»n*e  remainder 

to  zero. 
It  is 

+(«-'i9'r-^.-»e+«+2)i:'  ^^nrrlr'  (^•+^+«) 

+  («'^-y^e->e  +  /S  +  2)i:'*^^f±|f  (2e+6  +  i9) 
+  («'/?r-'*'e— e+r+2)i:>  yffilf  (2e  +  6  +  r) 
+  («-|9»r'^-V-'e  +  «+2)i:>^;t^' (2c+6  +  «) 

+  (a^^fS't'  -  •2e  +  2)}:'  /'."^/;,  (3*  +  6) 


+(«ry'«'e-»)i;+'+ 


e!e  +  3! 


.,  »»  +  e  +  2 


WIe  +  3! 

{(e  +  3)(*-2«)  +  »»} 


=0.  (30) 


5^  2'(ori.+.2'(«rr-«*«-*)'"^7r^ 


(31) 


jT  — e  +  2     Y     =0.2.4.6. ..2x 

—  2(a  +  *  +  c  +  rf+  *)'—  2»  +  2  \\\ 

(t)ri.+i   is  the    numerical    coefficient    of  /»i"»i^,_._,. 
oACidiitx  take  all  values  consistent  with  e^  <  e,  and 

ai  +  *i  +  ^i  +  </i  +  #1  =  the  constant  (a  +  *  +  ^  +  rf+  *  —  i). 


ft 

I 


I,   .! 


36 

(<^^f9*t—^y^  stands  for  the  numerical  coefficient  of 

(^J•)•/>J^)»/>iY)•/»i«l*/>i•>-') 

r  =  2x  —  a  (fli  +  *i  +  A  +  <^  +  *i  +  ») 

-  «  -  a,a  -  *i/J  -  Ar  -  «^*  —  (^»  +  0«- 

In  the  coefficient  (a'./S».r'«3*>«'0ii*— -r  ^  is  to  be  changed  to 
,_,_„_.  2.    When  *  -  2x  =  2  the  terms  that  must  be  added 

are  easily  recognized.  r  iHaiu-»)     t« 

For  an  example,  let  us  find  the  coefficient  of  Pl»lr!L*,   -    in 

Then 

|-  («+l)(o0ft''-'"+O  +  O  +  O+O+(</2)8r-""  — 


2x  —  lO .  2x  —  II 


»  +  I 


-(3*-4«-i2) 


+  ?_+i  [(o).((/)li-r."'+(o)i«(oP)«r-%">+(o)i*>(oP)li«-."> 
+  ...  +  (o)i5'-.i"(tf)w]  =  o. 

This  states  that 

n  +  I  times  the  coefficient  of  /1»i*iu"' 
+  twice  the  coefficient  of  P\PWlu*'^ 
^  a«-io.2«-ii  (y_^_  12)  times  the  coefficient  of  7n<?il».. 

+  ^+JL  times  a  number  of  terms  =  a 

Any  one  of  these  last  terms,  as  (o)^' Wi?-"."' . «  "S*  «S  «  f"" 
thus:  The  coefficient  of /Vi«.  times  the  coefficient  of /1»i«u+'« 

in  the  invariant  »i*i..  ,  -,„t^ 

As  another  example,  find  the  coefficient  of  /»J/>i»>  /^i"  '^-•«- 

Here 

2x  =  m  +  23,  tf  =  2,3  =  3.f  =  2, 

o  =  o,  i?  =  I,  t  =  3,  rr  =  1 .  3 . 5  .  •  •  a«  —  *7. 


tefficient  of 

r=jf  +  It  — 


m. 


r  —  d,9  —  (,ei+  i)«. 

;  is  to  be  changed  to 
\  that  must  be  added 

Qt  of  Pjoi^r."'.    In 

?  =  6, 

y  sz  2*  —  12  —  ff. 


10)  Jl 

3 

-  12) 


coefficient  of  71<?i^».. 


JT."' ,  is  written  in  full 
Efficient  of /1#i*iu+'. 


of/»J/>i»>*/'i*''^--- 


,  5  . . .  2x  —  17' 


27 

Then 

^^-^  (o^i'3')i:'  +  (o^»*3 . 5)i:'  (2 . 6  +  o) 
3 

+  r4i(3-6  +  0(<^i'3.6)J:' 

+  ^,  (2.6  +  3)(o^i'8)i:' 

+  HLZl^  (6i  -  lox  -  23)(6'i'3X+-^* 

Mt  I  Ol 

+  ^[(3)i"  {(o»i»3)&ri»+(|)c(o^i'3)l?!-. 
+  (rfi«2)ffL.}  +  (3)S{(<^i*3)l!?rr 

+  ^A)  C(o«i»3)ir-."  +  (f)  C(6'i'2)irrr 

+  (-^)  C(o*i3)irr."  +  [■^)c(.c^i'2)&=i> 
+  (A)  c(o'i*)irr.«  f  (-J)  C(tf3)i?'-. 
+  (-2-)  C(o*i2)irL.  +  (f )  C(o»i*)i:L,} 
+  (3)8'  { i^i*3)Szii  4-  (^)  C  (o»i»3)i?rJ| 
+  (f )  C(rfi'2)i?rj|  +  (^)  C(o*i3)i?rA' 

+(f )  C(tf3)i?-a  +  (-^)  C(o«i2)tfzS 

+  ("f )  c  (tf  lOirr  ji  +  (f )  c  (o^2)&=ii 

+  (-5-)  C(o*0&r}i>  (^)  C(o*i)!i-u} 

+ ■ 

+ 

+ 

(3)r--u»'  {(o^i)'*'  +  (oT'}  +  (3)ar-»i'»  {(o*)}] 


•  =o.  (33) 


H 


a8 
In  this  r  varies,  being  =y\-it  —  m  always,  and  C  also 
varies.  The  term  (3)?4(|-)(o^2)&-"  ^ea^  t^*  coefficient  of 
/»i"<>i*lu  times  the  coefficient  of  /»JPi'«i*l.*Vu  in  the  invariant 
^riu  multiplied  by  (-2-)  C-  r  =  m  -  5  +  9  -  w  =  4.  and  C 
is  the  numerical  coefficient  of  P\PfP^^  in  * 


^(PSmand(^)=;^ 


1' 


lilt 


Thus  every  term  in  the  invariant  #,  has  been  considered,  and 
by  (23),  (24)  and  (27)  every  coefficient  has  been  expressed  by 
simple  formulae  in  terms  of  *  and  n  excepting  those  represented 
by  (30),  and  they  are  expressed  in  terms  of  preceding  coefficients. 


xrays,  and  C  >1bo 

the  coefficient  of 

•!n«  in  the  invariant 

)  —  m  =  4i  and  C 

• 

rn- 

een  considered,  and 

been  expressed  by 

ig  those  represented 

■eceding  coefficients. 


Section  III. 

Associate  Equations  and  Associate  Variables. 

In  the  memoir  previously  referred  to,  Mr.  Forsyth  shows 
that  in  connection  with  any  differential  equation  Ai  of  order 
n  there  are  n  —  2  other  equations,  A,,  A,,  A^, . . .  A,^i,  whose 
variables  are  formed  as  follows :  Let  ih,  »,,«,,...«.  be  solu- 
tions of  ^],  then  if  we  take  any  two  «a,  »^,  the  determinant 

I  uxu^ 

!<< 

is  a  solution  of  At.    Generally  if  we  take  any  x  of  the  «'s  and 
form  a  determinant 


Uy  .    .    .    U, 

l^f  •        •       •       **» 


«ir-«  »],«-"  <-** 


=  ««, 


where  a,  /9,  ^  . . .  x  are  any  x  of  the  numbers  i,  2,  3  . . .  n,  then 
a.  will  be  a  solution  of  A,.    As  there  are  f — j  combinations  of 

n  tiiingB  X  at  a  time,  there  will  he  [— ]  variables  a.  satisfying  an 

equation  A,  of  order  ( -^j.    A,  will  be  called  the  (x  —  i)th 

associate  equation,  and  the  variables  a.  the  (x  —  i)th  associate 
variables.  These  variables  a«  are  particular  and  linearly  inde- 
pendent solutions  of  ^..  v4,_i  is  the  Lagrangian  adjoint  equa- 
tion. a«  may  be  written  (<^/' . . .  »<"~")f  or,  as  we  are  not 
concerned  with  which  suffixes  are  taken,  0123  ...  (x  —  i),  then 


5"  ...  n  —  i'"-*')  or  01234  ...(«  —  2), 


30 
a.=  («/9')  =  di,    a«  =  (a/8'r"0=oi23. 
The  number  of  these  «  (-^)  • 

«._i=("3  4   5  

while  (i2'3"4"'  •  •  • »»""')  °f  '234  •  •  •  (»  -  >) 

is  the  non.vanishing  constant  J.    To  illustrate  what  follows  I 

shall  first  take  a  particular  case,  n  =  5.    Then  i4i  will  be 

«">  +  lof ,»"  +  5f  4«'  +  ffi«  =  o-  (34a) 

«.  «..«..««.«.  are  the  five  independent  solutions ;  then  «,=oi. 
o  and  I  being  the  differential  indices  of  the  diagonal  of  the 
determinant  formed  with  any  two  of  the  u'a  and  their  first  de- 
rivatives, then 

^  =  fli   =02, 

Oi'   =03  +  I2,_ 

a',"  =  04 jf  2.I3,_      _ 
fl^  =  3. 14 +  2. 23  + OS- 
Substituting  for  u"  in  05  its  value  from  (35), 

«i«)  =  3 .  14  +  2 .  23  —  io<p,62  —  5f  .01, 

ar  +  lOfiPS  +  S*"*®!  =  3 •  14  +  2 .  23  =  J4,  say. 

Difierenriating,  __  _  _ 

5 .  24  +  3 .  I5J=  ^.  =  5i:*4  -.3  •  lo^'."  +  3<P.oi, 

Ji-3«».««  =  J»=5-24-30f«i^         _^  _ 

il=  5.34  -  30(^1" +  W)  + 5- 25_         _ 

=  5 .  34  -  30  (fii2  +  f,i3)  +  5  {5fi^2  +  *'»o2}> 
5i  _  5^,«i  =  5 .  34  +  (25«^*  -  3^.)  "  -  3PV»]Z=  '*'  say- 
ii  =  (25fi  -  SOfi')  "  +  (25f « -  30W 13  _ 

—  30f,Ci4  +  ^)  +  5-35 
=  (25fi  -  30p'i')  la  +  (50v»*^6o<pi)  13 

-3o^.(r4+^^^|^)+5^.K-r2) 

+  aSfiC**— 3'H).    _ 
y.~io?.5.-5f.«i'  =  -(5f.-25?'*  +  30fi')i2  _ 

+  f50f4-6ofi)  13-60^,14- 


!3- 


.(«— 2), 
0 

rhat  follows  I 
,  will  be 

(34a) 

;  then«i=oi. 
agonal  of  the 
their  first  de- 


*,  say. 

i5_         _ 
f^2  +  f.02}, 

,13  =  *••  say- 

?i)i3  _ 

+  23) +  5- 35 
pi)  13 

+  5V.(«i'-") 


kI)  13  —  60V.I4- 


iV^Fif  ■ii^fPif^«RW/?lvs- 


31 

Let 

-YS  -  (s^P.  -  25^«  +  30fl').   >'=(50f«-6o^i),  Z=-6o?.. 
and        '  , 

4  —  tOftSi  —  5f  »«i  =  h' 
Then 

J.  =  -X'i2  +  yi3  +  ^14,  (35) 

y,  =  A"i2  +  (A'+  r)i3  +  (K+Z')i4+  r23  +  Z(24  +  T5) 


'  —  "7  **  —  -f  (4  +  2f,at)  J 

y.  =.  [x"  +  2  ^)  12  +  (2^'  +  r '+ 1')  r3 
+  (-^+^+'?")H  +  (-^  +  n23 
+  (-^'  -  ^(M  + 15) 


(36) 


(--D 


(Ji  +  2?',fl,). 


Let 


,       /v/.  ^     ^^'       YZ\  — 


WMaaMMi 


3» 


-\-^    +     5     ^      »5  lo  30 

Now  we  have  four  equations,  (35).  (36).  (37).  (38).  by  which 
(12),  (13)  and  (14)  can  be  eliminated,  leaving 


(38) 


7* 

*'  ~5 30  ' 

-«  +  5     10 


y, 

y  +  x. 


r-^ 


r'  +  2A"  +  ^.      Z"-Y'^\X 


*Wl 


30 


r"+3A"'+ 


zr-i 

3 

ZY 

30  - 


rz'"- !>:::_  3^1 


=0. 


2 

z» 

30  J 

(39) 


an  equation  in  «.,  its  derivatives,  and  ^^f^^J^'^^f,^?^^ 
the  a)efficient8  of  (34a).    It  »  <>f  *»>«  ««"*  ^"^^'"^  *«"*  ^*"**'' 


iMlllliil 


^ 


12 


"  +  2X' 


.) 


.     (38) 


14 

;8),  by  which 


_  21 

3 

'  _  y  -  -  A" 

2  2 

30-1 

(39) 

derived  from 
:r  and  linear, 


^»» 


^ti 


^ti 


'■mi 


33 

and  is  the  first  associate  of  (34a).    To  obtain  the  second  asso- 
ciate, let  uf  represent  the  second  associate  variables.    Then 

w      =6l2, 

u/   =013, 
«^'  =  014  +  023, 
a^"  =  2 .  024  +  123  —  ioy>,  012, 
«^"  +  lofjzw  =  T,  =  2  .  024  +  123, 

rj  =  3 .  124  +  2 .  634  +  2 .  5^4  0I2, 
tJ  —  iof«ze>  =  3. 124  +  2. 034  =  T«,  aay, 
^«  =  5  •  '34  +  3 .  125  +  2  .  035, 
tJ  +  3f»»'  —  lOf^ie/  =  5 .  134  +  20<P,  023  =  T,,  say, 
^»  =  5  •  234  +  6oy,  123  +  2oyl  023 

_^  ■—  SVtit/  +  loy.r,, 

^•+  5f.«^  —  lOf.T,  =  5 .  234  +  6oiP.  123  +  2oyi  023  =  r,. 

Proceeding  thus,  four  equations  are  obtained  from  which 
024,  023  and  124  can  be  eliminated.    The  result  is 


A'{'  + 
"A'{''+ 


ZxZ\     y,z. 


4^.Z{' 


30  • 

z.rh 


5         10 
30 


=0, 


n  +  Xu 

z,z\-^ 

3 

z.r. 
30  J 


l'l"  +  3A'i'  + 


7/ ^ 

Z'i  -Y[  —  ^ 
2 

"^w_  Jl  yii_  3X'r\ 

'    2^»    ~r 
_z\ 

30  J 


where  JT,  =  5^?,  —  20<»J',  K  =  50^?^  —  140^; ,  Z  =  6of,. 
(40)  is  also  of  the  tenth  order  and  linear. 
The  third  associate  is  the  adjoint  equation.    It  is 


(40) 


34 

The  first  associate  of  this  adjoint  equation  may  be  obtained 
from  (39)  by  writing  in  it 

—  Vt  for  v%> 
5^4  —  20f  i  for  5^P4 . 

A  little  examination  will  show  that  these  transformations  among 
the  coefficients,  which  change  A,  into  A  and  A,  mio  A,,  also 
transforms  A,  into  A,  and  A,  into  ^,,  and  in  particular, 
s^  St,  s„  Su,  X,  VandZ 

into  T„T„T.,T,„X,,  K.andZ, 

respectively  and  vice  versa.    Then  for  the  quintic  at  least  it 

follows  that  the  rth  associate  of  an  equation  is  the^h  associate 

of  the  adjoint  equation  when 

Preparatory  to  extending  this  theorem  to  the  flth.c,  it  will  be 
well  to  consider  it  in  a  different  way. 

l(a,A,  represent  the  first  associate  variable  of  the  third  asso- 
ciate equation,  and  a.^.  the  (r-  Ost  associate  variable  of  the 
(i  —  i)st  associate  equation,  then 


.        1  (I2'3"4"')(.-6'S"9"') 


If  »  =  5,  then  6.  9,  8  will  be  2,  4.  3.  say,  and  the  above 

'^'°'"''         (23'4")(i5'2"3"'4'^)  =  --.(^3'4").    _ 

where  a,  is  the  non-vanishing  constant.  Thena,-*^^  =  CotA^,  C 

is  a  constant.    Take  «  =  6.    ^  is  the  adjoint    Then 

(I2'3"4"'5''),    (12'3"4"'6").    (I2'3"5"'6") 
12'3"4'"54.  (12'3"4"'6'7.  (I2'3"5"'6'^)' 

(12'3'V"5'T.  ("'3"4"'6")".  (i?'3"5"'6'T 

=  fl;(i2'3")   or   «i^i^*» 


a,/4. 


35 


obtained 


ins  among 
o  j4i ,  also 
ar. 


at  least  it 
ii  associate 

(42) 
:,  it  will  be 

third  asso- 
ible  of  the 


^5'6"8"'9") 
the  above 


=  Ca,Au  C 
en 

'6") 
'6")" 


IS 


then 

a,/4,  =  CotAi, 
where  C=  the  constant  J'.    The  general  theorem 

for  all  values  of  x  and  k  for  which  x  +  -l=  »;  that  is,  the  *-  i)at 
assocate  variable  of  the  adjoint  equation  is  a  constant  multiple 
of  the  (A-i)st  associate  variables  of  the  original  equation  when 

««/^,_,  is 

C23'4"5"' . .  .  »"-").  (ISV'S'"  .  -  «-  I'— ',  «(— )) 

(I3V'5"'6"  . .  .  »-  ,<— ),  «<-«),.. . 

(I2'3"  .  .  .  x_  x'«-«',  *  +  i(«-i)  ^  ."  J,(.-i)) 

(23'4"5"' . . .  «'—')'.  dsV's'"  ...»  -  i<— >,  «'—))'. 

( )'     .       /  '        y 

(23'4"5"' . . .  »'—•)''.  (13V5'"  .  . .  «"-')"'. 


(23'4"5"' . . .  «'— ')'«-", ... 

(I2'3"  .  .  .  ,  _  i(«-«»,  ,  +  i(.-.)  ^  ^  ^  „(.-,,)„-„ 

This  is  a  determinant  of  order  *.  In  the  third  and  lower 
rows  each  constituent  equals  the  sum  of  a  number  of  terms  all 
but  one  of  which  will  contain  «<•'.  and  substituting  for  thii  its 

mulLrnf'^'  '•?•'"'"*"'  equation,  the  terms  arf  seento^L 
multiples  of  preceding  rows  and  may  be  omitted.    Each  con- 

deSJis  ir ''""  '"*  "'""^  °^  1'  ""^  '^^  --i"««»« 

I'— ',  2'-»,  3(-«),  4(.-.)...,,.-., 
I'"-',  a'—',  3'—'.  4'—' . . .  «'"-' 


i'"~"'.  a'— ',  3'"-'.  4'"-"...«'"-«' 


agi 


3« 

Having  found  a  proof  showing  that  <i,/^._,  =  fl.-.'^.J""' 
was  not,  in  general,  true,  I  used  it  for  the  case  when  v=i, 
when  it  is  true  that  a.^._.  =  a._.^.J— .  But  this  follows 
immediately  from  Section  6,  Chapter  V,  of  Determinants,  by 
R.  F.  Scott.  Then  we  conclude  that  for  all  values  of  *  the 
(x  —  i)8t  associate  variable  of  an  equation  is  a  constant  multiple 
of  the  («  —  X  —  i)8t  associate  variable  of  its  adjoint  equa- 
tion. ^  ^  (45) 

When  yii  is  self-adjoint,  A,-i  =  A,  and  then 

or  all  equations  of  complementary  rank  associate  to  a  self-adjoint 
equation  are  equal.  (^o) 

The  associate  equations  A»  and  /f,-,  are  said  to  be  of  com- 
plementary rank. 

The  question  arises,  does  this  hold  for  other  associate  equa- 
tions of  complementary  rank,  i.  e.  for  any  equation  does 

Turning  to  equations  (39)  and  (40),  make 
y,  =0    and    fi  =  5ff«» 
then  (39)  reduces  to  an  equation  of  the  ninth  order,  there  being 
a  linear  relation  between  the  a's.    But  /«,  or  (40)  does  not 

reduce. 

a,/4,  is  now  a  non-vaniahing  constant  and  cannot  be  a  solution 
ofAr    Therefore  a,^,  does  not  equal  ai.4,.  (47) 


i 


.     I 


\ 


«    ^ 


Sech'0/i  IV. 

Conditions  for  the  Sblf-Adjointness  of  Differential 

Equations. 

Any  equation  is  sdf-adjoint  when  its  invariants  with  odd 
suffix  vanish. 

Let  r  be  the  order  of  the  equation.  The  relations  which 
exist  between  the  coefficients  are 


(47a) 


+  (^)/'iV+...      «  =  i.2.3....r  J 

These  relations  follow  from  those  given  by  Dr.  Craig  in  his 
treatise,  pp.  490-493.  For  example,  take  the  sextic  (y),  p.  491, 
and  (r)',  p.  492.    In  order  that  it  may  be  sdf-adjoint, 

P.  =  P^3P,  +  6P'>, 
or  generally, 

(-.)■/>... =--r(-»-(^)/'fi.-.. 

If  the  equation  had  been  written  with  binomial  coefficients 
this  would  become 

If  we  call  6  —  X,  f»  and  divide  (— j  it  becomes 
(-  lyPm  =  /*.  ~  mPi,_^  +  ~,  etc. 

It  is  not  difficult  to  see  that  this  will  hold  for  any  equation. 


38 
First,  let  »  be  odd,  then 

o=2/'.-«/'i_.  +  (^)n'-.-(-|)/'l"-.+  -.etc.    (48) 

2^.=  2/>.-,./>._.  +  ^^(-f  )/>:'_, 

_n-2\2n-5\(n\ 

2«  — 3\2/  2«  — 3\2/  L 

•Thus  it  is  seen  that  (48)  —  2^,  contains  neither  P,  nor  Pi-i, 
and  that  (48)  -  2©,—  (— ]  ~~z-  ^--«  '*  without  the  first  two 
pair  of  terms  m  Pn,  P'n-i,  Pll-t,  Pi!'-,,  and  from 

(4») -».  -  (t)  ^3  «- (f)M(i^)  "'-• 

the  first  three  pairs  of  terms  disappear.  By  subtracting  certain 
multiples  of  the  invariants  and  their  derivatives  from  (48)  the 
terms  continue  to  disappear  in  pairs.     The  multiplier  of  ^il'iv 

would  be  2  (^)(^)(,^  _';_,)  =  ^K>  say. 
From  what  precedes,  especially  (22)  and  (23),  we  know 
the  coefficient  of  Pj^„  in  (48)  is  (j~\ , 
the  coefficient  of /*i'!L'i«  in  2i%^,  is 

the  coefficient  of  Pi'l'„  in  2M^J,'_,  is 


*"««■ 


.etc.    (48) 


+  ... 


.4  +  • .  •    • 

,  nor  PUi, 
lie  first  two 

5)    -* 

ting  certain 
)m  (48) the 
ierof^i'l'^ 


know 


•  •        • 

•  •        • 


<\ 


39 


the  coefficient  of  P<«'«  in  2M„ev^„  is 
'^' U  -  2^K^;r^ri^^\2n-^^-^  =  M,C,  say. 


It  will  now  be  shown  that 


M„C„ 


Let 
then 


(-^^)  =  i>^"  "■'=-^7zy 


i%C_ 


(i) 


=  t^aCa, 


m  r    —  "  —  I  !  2«  —  2X  —  2  !  2M  —  1  .  2X  ! 

"•oto  — j— i — r 

2»\n  —  2X—l  !  2M  —  I  !  ' 

Witf,  =  »— ii2"-2«-4!2«-5.2;t! 

2  !  2«  —  3  !  2X  —  2  !  «  —  2x  —  I  1   ' 

generally 

m^„  =  .' Jl")    «— I !  «-2 .  «-3  . . .  n—2x .  2«-4<r-i 

\  2cr  /  2«— 2<r— I  .  2«— 2(r— 2  .  .  .  2»— 2<r— 2x— I  * 

When  «  =  I,  m,c„  has  a  zero  factor  in  the  numerator  for  all 
values  of  <r  except  «r  =  a    The  series  reduces  to 


tftoCo  = 


—  2x  —  I  ! 

—  2x  —  I  ! 


=  I. 


For  n  =  2  the  series  has  no  zero  factor,  if  ir  =  o  or  i,  and 
reduces  to 

—  2*  —  2 1 3  2x  —  2 !  2x  .  2z  —  I 

+       ; , _     ,^ 


2«  —  I  !  2 


3.2.I.2X— 3 

Similarly  for  «  =  3,  4,  5. 

For  «  =  2x  the  series  is  m,c,  _  2x  —  i !  2 

For  n=-  2x  —  I  the  series  is 


2x  —  I  I  2 


—  W«<',  +  tftK^iCn^i  ■=: 


2X  —  2  !  2X  ! 


=  I. 


2x 


2*  —  2  !  2*  —  I  !  2         2x  —  3 !  2 


2! 
n-r  =  I. 


iwiliMtt 


40 

For  «  =  X  —  I, 

2x  !  2x  —  3    ,       ,^,  X  — 2l«  +  ll 


tttnPo 


2X  —  3 !  3 ! 
2X  !  2x  — 

2x  —  2 ! : 

_  2x  1  2x  !  2«  —  1 1 

*«•''•  -  " 4!2x-4!7!2x-7 


2x 


«.^  ,        2x!2x-7.2x! 

*«''^'-*'2!2x-2!2x-5!5l^       ^^' 

(- 1)-. 


•       •       • 


•       •        •       • 


•         ■         •         • 


•         •        • 


2x  !  2x  !  2x  —  9  ,         .,_, 

.,.....-,  =  «(f)(^)^x- 5  (-i)-S 

»io«'o  —  »«,_i^,_»  +•  i^Ci  —  »»,_|f«_,  +  ntjCt 1- ,  etc. 

forms  a  series  which  is  equal  to  unity.    This  is  seen  by  taking 
the  coefficient  of  y+*  from  each  member  of  the  equation  in 

which  (i  —yy  4-  (i  +  J')*"  is  written  equal  to  its  expansion 

(-^)— ^-(Ty-(lV-(Tk-(fy 

+ . . . +(f  )y-*  -  ( j)y—  +  ( j)y--2«y-+y". 

+  ...  +  8(|:)y  +  7(f)y  +  ...  +  4(^^)y  +  ... 

The  coefficient  ofy  *^*  in  the  product  of  the  right  members 


IS 


{^(?)-=(t)(?)-*(?X?)     -'(yXi)^-- 


i» 


} 


1.^ 


mm 


which  is  the  series  [£«££-'(.  ,)..     The  coefficient  ofy +  •  in 


2*(i  _j,)(i  — y)'«-«  is 


Therefore 

»=« 

.io*"'''=''  (49) 

Then  for  all  velues  of  «  in  like  manner  the  same  result  will 
follow,  and  thus  the  coefficient  of  />i«!'^  in  (48) 

=  ^Mfin  +  2Jif,ei_,  +  2Af,e]r_,  +  .,.  +  2jif,ei,*i>^,   (50) 

of  ^.!.'°K*'-*"*  °^^----' '"  th«J««  series  is  found  from  that 
of  /»<«'„  by  giving  a  the  same  values  and  changing  2x  to  a«  +  i. 

and  therefore  this  also  equals  (     "     ] 

ouH"-  '^''*^  "     /r°'  *•**  8^*"*'"**  ""^'^t'O"  between  the  coeffi- 
cients  IS  expressed  by 

.re^        '[.(50 
thitVcxV^  '™'^  *""  ?*'  *^*  *''"°  «  ^  ^^J**'  't  "^y  be  shown 

(50  =  #._.+  ^.ei'_.  +  JV^eiL,  +  . . .  +  AT _,»i.-.,.  (32) 

wh^n  V"  •  '"^*"»°*«  '"  (50)  and  (52)  have  odd  suffixes.  Then 
when  the  invariants  with  odd  suffixes  vanish  (48)  equals  zero! 


''*^— ■"*"""T'W[n«i»iiiiB[w[mrw»fiiii 


■>»Mfta»fctiruw«iM<c*y^Maittiaw'.»aMflMa :  QUN^b^rMdhUSi^iM'a 


42 

and  also  (51)  equals  zero,  and  the  conditions  for  self-adjointness 
are  satisfied,  and  the  proposition  with  which  this  section  begins 
is  established. 

It  is  to  be  noticed,  however,  that  an  equation  may  be  self- 
adjoint  when  its  invariants  with  odd  suffix  do  not  vanish,  but 
satisfy  the  linear  relation  expressed  by  equating  the  right  mem- 
bers of  (50)  and  (52)  to  zero,  which  is  equivalent  to  saying  that 
(47a)  and  (57)  are  satisfied. 


U 


^ 


««ii»i*.-,«(#fi'.«ssa«asB»»«*SBS»«'  -«?»!^i 


ointness 
(1  begins 

be  self- 
lish,  but 
bt  mem- 
ing  that 


\> 


U 


Biographical. 

George  Frederic  Metzler,  the  son  of  George  Frederic  and 
Nancy  Ann  (Shannon)  Metzler,  was  born  July  17,  1853,  at 
Westbrook,  County  of  Frontenac,  Ont.,  Canada.  His  early 
education  was  received  at  the  Odessa  public  schools  and  at 
different  high  schools.  His  collegiate  education  was  received 
at.Albert  College,  Belleville,  Ont.  (now  consolidated  with  Vic- 
toria  College  and  federated  with  Toronto  College  in  Toronto 
University).    At  Albert  College  he  took  the  degree  A.  B.  in 

1880,  and  the  degree  M.  A.  in  1883.  He  has  taught  going  on 
two  years  in  public  schools,  two  years  in  high  school,  one  year 
as  head-master,  and  was  called  to  teach  in  Albert  College  in 

1881.  He  entered  Johns  Hopkins  University  October,  1884, 
remained  one  session,  entered  again  1887.  He  taught  in  Ma- 
rietta College,  Ohio,  1889-90.  The  present  year  he  spent  in 
Baltimore  preparing  for  the  degree  Ph.  D.  His  studies  have 
been  in  mathematics,  astronomy  and  physics. 

Baltimobb,  Md.,  1890-91. 


^j^,U...iiiii(|]|HiHi., 


